MAISRN Mathematical Analysis2090-4665International Scholarly Research Network94574110.5402/2012/945741945741Research ArticleBoundedness and Compactness of the Mean Operator Matrix on Weighted Hardy SpacesYousefiBahmann1PazoukiEbrahim1BerettaE.KravchenkoV.1Department of MathematicsPayame Noor UniversityP.O. Box 19395-3697, TehranIranpnu.ac.ir2012852012201205012012160220122012Copyright © 2012 Bahmann Yousefi and Ebrahim Pazouki.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the boundedness and the compactness of the mean operator matrix acting on the weighted Hardy spaces.

1. Introduction

First in the following, we generalize the definitions coming in . Let β={β(n)} be a sequence of positive numbers with β(0)=1 and 1<p<. We consider the space of sequences f={f̂(n)}n=0 such thatfp=fβp=n=0|f̂(n)|pβ(n)p<. The notationf(z)=n=0f̂(n)zn will be used whether or not the series converges for any value of z. These are called formal power series and the set of such series is denoted by Hp(β). Let f̂k(n)=δk(n). So fk(z)=zk and then {fk}k is a basis such that fk=β(k). Recall that Hp(β) is a reflexive Banach space with norm ·β and the dual of Hp(β) is Hq(βp/q) where 1/p+1/q=1 and βp/q={β(n)p/q} . For some other sources on this topic see .

The study of weighted Hardy spaces lies at the interface of analytic function theory and operator theory. As a part of operator theory, research on weighted Hardy spaces is of fairly recent origin, dating back to valuable work of Allen Shields  in the mid- 1970s. The mean operator matrix has been the focus of attention for several decades and many of its properties have been studied. Some of basic and useful works in this area are due to Browein et al. , which are pretty large works that contain a number of interesting results and indeed they are mainly of auxiliary nature. Also, some properties of mean operator matrices have been studied recently by Lashkaripour on weighted sequence spaces . In this paper, we have given conditions under which the mean operator matrix is bounded and compact as an operator acting on weighted Hardy spaces. More details of our works are as follows: the idea of Theorem 2.6 comes from . In Theorem 2.9, we extend the method used in [20, Theorem 1.2] to show the boundedness of the mean operator matrix acting on the weighted Hardy spaces. Some inequalities are useful to find a bound for the mean operator matrix acting on weighted Hardy spaces . For example the inequality proved in [26, Theorem 8] is used in the proof of Theorem 2.11.

2. Main Results

In this section we define an operator acting on Hp(β) and then we will investigate its boundedness and compactness on Hp(β).

Definition 2.1.

Let {an} be a sequence of positive numbers and define An=i=0naiβ(i)p. The mean operator matrix associated with the sequence {an} is represented by the matrix A=[ank]n,k and is defined by ank={akβ(n)pAn,0kn,0,k>n.

From now on, by A we denote the mean operator matrix associated with the fixed sequence {an} as in Definition 2.1.

Theorem 2.2 (see [<xref ref-type="bibr" rid="B26">12</xref>, Theorem 1]).

If 0<anan+1 for all integers n0, then A is a bounded operator on Hp(β).

Theorem 2.3 (see [<xref ref-type="bibr" rid="B26">12</xref>, Theorem 2]).

Let 1/p+1/q=1 and bn>0 for n=0,1, If M1=supn0k=0nakβ(n)p+1Anβ(k)(bkbn)1/p<,M2=supk0n=kakβ(n)p+1Anβ(k)(bnbk)1/q<, then A=[ank]n,k is a bounded operator on Hp(β) and AM11/qM21/p.

Recall that if an,bn are two positive sequences, by an~bn, we mean that an/bn1 whenever n. Also, we write an=o(bn), if an/bn0 as n.

Corollary 2.4.

Let limnnan/An be finite and 1/p+1/q=1. If supn0k=0nakβ(n)p+1nanβ(k)(bkbn)1/p<,supk0n=kakβ(n)p+1nanβ(k)(bnbk)1/q<, then A is a bounded operator on Hp(β).

Proof.

Put limnnan/An=β. Then nan/β~An and so k=0nakβ(n)p+1Anβ(k)(bkbn)1/p~βk=0nakβ(n)p+1nanβ(k)(bkbn)1/pas  n,n=kakβ(n)p+1Anβ(k)(bnbk)1/q~βn=kakβ(n)p+1nanβ(k)(bnbk)1/qas  k. On the other hand supn0k=0nakβ(n)p+1nanβ(k)(bkbn)1/p<,supk0n=kakβ(n)p+1nanβ(k)(bnbk)1/q<, thus Theorem 2.3 implies that A is a bounded operator on Hp(β).

Lemma 2.5.

Suppose that ncan/β(n) is eventually increasing when the constant c>1-γ, and eventually decreasing when c<1-γ. Let S1(n)=1nk=1nakβ(n)anβ(k)(kn)-1/p,S2(k)=k1/qn=kakβ(n)anβ(k)1n1/(q+1). If γ>1/p, then limnS1(n)=limkS2(k)=1/(γ-1/p).

Proof.

Let 1/p+1/q=1 and c2<1-γ<c1<1. Then in either case there is a positive integer N such that (kn)-c2<akβ(n)anβ(k)<(kn)-c1 for Nkn. Suppose first that γ>1/p, then limnn1-1/panβ(n)= and hence limn1nk=1N-1akβ(n)anβ(k)(kn)-1/p=0. Therefore limnsupS1(n)limn1nk=Nn(kn)-c1-1/p=01x-c1-1/pdx. By calculus integral we get 01x-c-1/pdx=11-c-1/p;c1q, and so limninfS1(n)limn1nk=Nn(kn)-c2-δ=01x-c2-1/pdx=11-c2-1/p. Letting c11-γ from the right and c21-γ from the left, we have limnS1(n)=1γ-1/p. Also note that limksupS2(k)limkk1/qn=k(kn)-c11n1/q+1,limkk1/qn=k(kn)-c11n1/q+1=11/q-c1. If c11-γ, then 1/q-c1γ-1/p and similarly we get limkinfS2(k)limkk1/q-c2n=k(1n)1-1/q-c2,limkk1/q-c2n=k(1n)1-1/q-c2=11/q-c2. If c21-γ, then 1/q-c2γ-1/p. This completes the proof.

Theorem 2.6.

Let limnnanβ(n)p/An=γ, ncanβ(n)p be eventually monotonic for any constant c, and {β(n)} be bounded. Then A is a bounded operator if 1/γ<p.

Proof.

Let δn=nanβ(n)p/An and suppose first that 0γ<. Then n(log(An)-log(An-1))=-nlog(1-δnn)γ as n, and hence log(An)-log(A1)=-nk=2nlog(1-δkk)=ϵnlogn, where ϵnγ. Consequently An=A1nϵn. Now suppose that γ=, then for n2, log(An)-log(An-1)=-log(1-δnn)δnn since δn. If M>0, then there is N1 such that δnM+1 for all nN1.

Without loss of the generality suppose that there is a positive real number a>0 such that δn>a for nN1. Note that k=2N11k=logN1+c+o(1)-1. If n>N1, then k=2n1k=logn+c+o(1)-1,k=N1+1n1k=logn-logN1. Also, k=2nδk/klogna(k=2N11/k)+(M+1)(k=N1+1n1/k)logn,k=2N11/k+k=N1+1n1/klognM1+1(logn-logN1)+a(logN1+c+o(1)-1)logn=M1+1+(a-M1-1)logN1+a(c+o(1)-1)logn, for large amount of n last equality greater than M1. Hence logAnk=2nδkk=γnlogn, where γn. It follows that, for any real number c, ncAn=nc+γn. Since nc-1An~1γncanβ(n)p, thus ncanβ(n)p is eventually increasing for c>1-γ, and eventually decreasing for c<1-γ. But {β(n)}n is bounded, so there are M1,M2>0 such that M1<β(n)<M2, and ncanβ(n)=ncanβ(n)pβ(n)p+1,ncanβ(n)pβ(n)p+1ncanβ(n)pM1p+1. This implies that ncan/β(n) is eventually increasing for c>1-γ. Similarly ncan/β(n) is eventually decreasing for c<1-γ. Thus k=1nakβ(n)p+1Anβ(k)(kn)-1/p~γnk=1nakβ(n)p+1nanβ(k)(kn)-1/p. By Lemma 2.5γnk=1nakβ(n)p+1nanβ(k)(kn)-1/p is bounded and so k=1nakβ(n)p+1Anβ(k)(kn)-1/p is bounded. We can see that n=kakβ(n)p+1Anβ(k)(kn)1/q is also bounded. Now by Theorem 2.3, A is a bounded operator and so the proof is complete.

Lemma 2.7.

Let {an},{tn} be nonnegative sequences with t-1=0. Then for all n one has k=0n(tkak){max0kn(1n-k+1j=Knaj)}(k=1n(n-k+1)(tk-tk-1)++t0(n+1)).

Proof.

Employing the summation by parts, we get k=0n(tkak)=k=0n(j=knaj)(tk-tk-1)k=0n(j=knaj1n-k+1)(tk-tk-1)+(n-k+1). So k=0n(tkak){max0kn(1n-k+1j=knaj)}(k=1n(n-k+1)(tk-tk-1)++t0(n+1)), and at this time the proof is complete.

Theorem 2.8 (see [<xref ref-type="bibr" rid="B6">26</xref>, Theorem 8]).

Let 1/p+1/q=1, {xn} be a positive sequence, then j=0max0ij(1j-i+1k=ijxk)pqp(k=0xkp).

Theorem 2.9.

Let {an} be a positive sequence and M3=supn0(k=1nn-k+1An(akβ(k)-ak-1β(k-1))+β(n)p+1+(n+1)a0Anβ(0)β(n)p+1) be finite. Then A is bounded and AM3q.

Proof.

Let f(z)=n=0f̂(n)znHp(β), thus A(f)(z)=n=0(k=0nakβ(n)pAnf̂(k))zn. By definition of ·β, we have n0β(n)p|k=0nakβ(n)pAnf̂(k)|pn0(k=0nakβ(n)p+1Anβ(k)|f̂(k)|β(k))p. In Lemma 2.7, consider tk=ak/β(k) and aj=|f̂(j)|β(j). Then n0(k=0nakβ(n)p+1Anβ(k)|f̂(k)|β(k))pn0{max0kn1n-k+1j=kn|f̂(j)|β(j)}p   ×(k=1nn-k+1An(akβ(k)-ak-1β(k-1))+β(n)p+1+(n+1)a0Anβ(0)β(n)p+1)p. Now, Theorem 2.8 implies that n0{max0kn(1n-k+1j=kn|f̂(j)|β(j))}pM3pM3pqpk=1|f̂(k)|pβ(k)p, and so we get AfM3qfβ for all fHp(β). Thus AB(Hp(β)) and indeed AM3q. This completes the proof.

Corollary 2.10.

Let 1/p+1/q=1, ak/β(k)ak-1/β(k-1) and M4=supn0k=0nakβ(n)p+1β(k)An<. Then A is a bounded operator on Hp(β) and AM4.

Proof.

Note that n0(k=0nakβ(n)p+1Anβ(k)|f̂(k)|β(k))pn0{max0kn1n-k+1j=kn|f̂(j)|β(j)}p(k=0nakβ(n)p+1β(k)An)p. Theorem 2.8 implies that n0{max0kn(1n-k+1j=kn|f̂(j)|β(j))}pM4pM4pqpk=1|f̂(k)|pβ(k)p, and so by Theorem 2.9 we obtain AfqM4fβ for all fHp(β). Thus AB(Hp(β)) and indeed AM4q. This completes the proof.

Now, we characterize compactness of subsets of Hp(β) and then we will investigate compactness of the mean operator matrix on Hp(β).

Theorem 2.11.

Let S be a nonempty subset of Hp(β). Then S is relatively compact if and only if the following hold:

there exists M>0, such that for all n=0f̂(n)znS, |f̂(i)β(i)|M for all i{0};

given ϵ>0, there is n0 such that n=n0|f̂(n)|pβ(n)p<ϵp for all n=0f̂(n)znS.

Proof.

Let S be relatively compact, thus there exist g1,,gkHp(β) such that Si=1kB(gi,1). For every f(z)=n=0f̂(n)znS, there is gi such that fB(gi,1). By Minkowski inequality we get n=0|f̂(n)|pβ(n)p[(n=0|f̂(n)-ĝi(n)|pβ(n)p)1/p+(n=0|ĝi(n)|pβ(n)p)1/p]p(f-gi+gi)p(1+gi)p(1+max{gi:i  =1,,k})p. Thus for every fS and n{0}, we get |f̂(n)β(n)|1+max{gi:  i=1,,k}. So (i) holds. Now suppose that ϵ is an arbitrary positive number. Since S is relatively compact, thus there exist h1,,hkHp(β) such that Si=1kB(hi,ϵ2). Since hiHp(β), there exists Ni such that n=Ni|ĥi(n)|pβ(n)p<ϵp2p for i=1,,k. Put N0=max{Ni:  i=1,,k}, and consider fS. Then there exists i{1,,k}, such that fB(hi,ϵ/2). Hence we get n=N0|f̂(n)|pβ(n)p[(n=N0|f̂(n)-ĥi(n)|pβ(n)p)1/p+(n=N0|ĥi(n)|pβ(n)p)1/p]p(f-hi+ϵ2)pϵp. So (ii) holds.

Conversely, assume that ϵ>0 be given and let (i) and (ii) hold. By condition (ii), there exists n0 such that n=n0|f̂(n)|pβ(n)p<ϵp2, for all fS. Let Mn0 be the closed linear span of the set {1,z,,zn0-1} in Hp(β). Consider n0 and Mn0 with norms (z1,,zn0)=(n=1n0|zn|pβ(n)p)1/p, for all (zi)i=1n0n0, and i=0n0-1aizi=(i=0n0-1|ai|pβ(i)p)1/p for all i=0n0-1aiziMn0. Define L:Mn0n0, by L(i=0n0-1aizi)=(a0,,an0-1). Clearly, we can see that L is a bounded linear operator. Now, consider the compact subset {(zi)i=1n0:i=1n0|zi|pβ(i)pn0Mp} in n0. Then we have {i=0n0-1f̂(i)zi:n=0f̂(n)znS}L-1{(zi)i=1n0:i=1n0|zi|pβ(i)pn0Mp}. Since L-1{(zi)i=1n0:i=1n0|zi|pβ(i)pn0Mp} is a compact subspace of Mn0, so there exist g1,,gkMn0 such that L-1{(zi)i=1n0:i=1n0|zi|pβ(i)pn0Mp}i=1kB(gi,ϵ21/p). Hence for every f{i=0n0-1f̂(i)zi:f(z)=n=0f̂(n)znS} there is i{1,,k} satisfying n=0n0-1|f̂(n)-ĝi(n)|pβ(n)pϵp2. Also, we have (f-giβ)pn=0n0-1|f̂(n)-ĝi(n)|pβ(n)p+n=n0|f̂(n)|pβ(n)pϵp2+ϵp2ϵp. Thus, S is relatively compact and so the proof is complete.

Theorem 2.12.

Let the mean matrix operator A be bounded on Hp(β), and limm(n=mβ(n)p2+pAnp)1/p(k=0m(akβ(k))q)1/q=0, where 1/p+1/q=1. Then A is a compact operator on Hp(β).

Proof.

Let BHp(β) be the closed unit ball of Hp(β). Define S=A(BHp(β)) and note that S is a bounded subset of Hp(β). Put rn=|f̂(n)|an, un=β(n)p2+p/Anp, vk=(β(k)/ak)p, and Em=(n=mun)1/p(k=0mvk1-q)1/q. Note that limmEm=0. So for every ϵ>0, there exists m0 such that Em<ϵ/(qp-1p)1/p for all mm0. Note that if f(z)=k=0f̂(k)zkBHp(β), then Af(z)=n=0(k=0nakβ(n)pf̂(k)An)znS. Since fβp1, we have n=m|Âf(n)|nβ(n)pn=mβ(n)p2+pAnp(k=0nak|f̂(k)|)p=n=mun(k=0nrk)pϵpk=0(rk)pvkϵpk=0|f̂(k)|pβ(k)pϵp. Thus by Theorem 2.11, S is compact and so the proof is complete.

ShieldsA. L.Weighted shift operators and analytic function theoryTopics in Operator Theory1974Providence, RI, USAAmer. Math. Soc.491280361899ZBL0303.47021YousefiB.On the space lp(β)Rendiconti del Circolo Matematico di Palermo. Serie II2000491115120175345610.1007/BF02904223ZBL0952.47027YousefiB.Unicellularity of the multiplication operator on Banach spaces of formal power seriesStudia Mathematica20011473201209185376810.4064/sm147-3-1ZBL0995.47020YousefiB.Bounded analytic structure of the Banach space of formal power seriesRendiconti del Circolo Matematico di Palermo. Serie II2002513403410194746310.1007/BF02871850ZBL1194.47035YousefiB.JahediS.Composition operators on Banach spaces of formal power seriesBollettino della Unione Matematica Italiana2003624814871988217ZBL1150.47014YousefiB.Strictly cyclic algebra of operators acting on Banach spaces Hp(β)Czechoslovak Mathematical Journal200454(129)126126610.1023/B:CMAJ.0000027266.18148.902040238ZBL1049.47033YousefiB.Composition operators on weighted Hardy spacesKyungpook Mathematical Journal20044433193242095415ZBL1085.47031YousefiB.DehghanY. N.Reflexivity on weighted Hardy spacesSoutheast Asian Bulletin of Mathematics20042835875932084748ZBL1081.47039YousefiB.On the eighteenth question of Allen ShieldsInternational Journal of Mathematics20051613742211567610.1142/S0129167X05002758ZBL1081.47038YousefiB.KashkuliA. I.Cyclicity and unicellularity of the differentiation operator on Banach spaces of formal power seriesMathematical Proceedings of the Royal Irish Academy2005105A11710.3318/PRIA.2005.105.1.12138718YousefiB.FarrokhiniaA.On the hereditarily hypercyclic operatorsJournal of the Korean Mathematical Society200643612191229226468010.4134/JKMS.2006.43.6.1219ZBL1114.47008YousefiB.BagheriL.Boundedness of an operator acting on spaces of formal power seriesInternational Journal of Applied Mathematics20072068218252358821ZBL1145.47026BorweinD.JakimovskiA.Matrix operators on lpThe Rocky Mountain Journal of Mathematics19799346347752874510.1216/RMJ-1979-9-3-463ZBL0427.47020BorweinD.Simple conditions for matrices to be bounded operators on lpCanadian Mathematical Bulletin19984111014BorweinD.GaoX.Matrix operators on lp to lqCanadian Mathematical Bulletin199437444845610.4153/CMB-1994-065-71303670ZBL0819.47039BorweinD.Weighted mean operators on lpCanadian Mathematical Bulletin. Bulletin Canadien de Mathématiques200043440641210.4153/CMB-2000-048-31793942ZBL1017.47026LashkaripourR.Weighted mean matrix on weighted sequence spacesWSEAS Transactions on Mathematics2004347897932119309ZBL1205.40009LashkaripourR.Lashkari@hamoon.usb.ac.irTranspose of the weighted mean matrix on weighted sequence spacesWSEAS Transactions on Mathematics200544380385LashkaripourR.ForoutanniaD.Inequalities involving upper bounds for certain matrix operatorsProceedings of the Indian Academy of Sciences. Mathematical Sciences20061163325336225600910.1007/BF02829749ZBL1118.47024LashkaripourR.ForoutanniaD.Extension of Hardy inequality on weighted sequence spacesJournal of Sciences. Islamic Republic of Iran20092021591662562708ZBL1177.26039HardyG. H.LittlewoodJ. E.PólyaG.Inequalities1952Cambridge, UKCambridge University Pressxii+3240046395CartlidgeJ. M.Weighted mean matrices as operators on lp, Ph.D. thesis1978Indiana UniversityKufnerA.MaligrandaL.PerssonL.-E.The Hardy Inequality, About Its History and Some Related Results2007Plzeň, Czech RepublicVydavatelsky Servis1622351524KufnerA.PerssonL.-E.Weighted Inequalities of Hardy Type2003River Edge, NJ, USAWorld Scientificxviii+3571982932BennettG.Some elementary inequalitiesThe Quarterly Journal of Mathematics. Oxford. Second Series19873815240142591622510.1093/qmath/38.4.401ZBL0649.26013HardyG. H.LittlewoodJ. E.A maximal theorem with function-theoretic applicationsActa Mathematica193054181116155530310.1007/BF02547518